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Full text of "Treatise On Analysis Vol-Ii"

412       XV    NORMED ALGEBRAS AND SPECTRAL THEORY

24.    Let E be a separable Hilbert space and H a positive self-adjoint operator ; then / -f A#
is invertible for all A > 0. For x e E and A > 0 let FA(x) = (A(/+ A/7)-1 'X\x). Show
that F*(x) is increasing as a function of A, and that it is bounded if and only if
x e H1/2(E). (Reduce to the case where H is a simple operator (15.11.3).)

25.    Let E0 be a real Hilbert space and E the Hilbert space obtained by extending the field
of scalars of E0 to C, so that every element of E is uniquely of the form x + iy with
x, y e Eo , and

(*' + iy' | x" + iy") = (*' | x") + (/ 1 /') + /(/ 1 x") - f(*' | y).

Show that every self-adjoint operator ffQ on E0 extends uniquely to a self-adjoint
operator H on E, having the same spectrum.

26.    With the notation of Problem 2 of Section 13.13, suppose that for each compact
subset K of X there exists a constant bk i> 0 such that

(B)                                              f \u\

JK

for all us tf. This condition implies condition (A) of Problem 2, Section 13.13, but is
not equivalent to (A).

(a) In Problem 5(b) of Section 6,6, suppose that X is compact and that the functions
fn are real-valued, bounded and measurable with respect to a positive measure ^ on X
and that they satisfy the condition

where ||/J = sup \fn(x)\. Show that the space & of functions which are /^-equivalent

* e X

to the functions belonging to the space denoted by E in the problem referred to, is
such that H = tfFlJf is a Hilbert space isomorphic to E, and satisfying condition
(B) above.

(b)    For every function /e &&(X, //,) with compact support K, there exists a function

U7 E & such that (Uf \ u) = \ufd\ji for all u e tf. The class of Uf is uniquely deter-

mined by the class of/, and we have (U'l < 6ic/2N2(/). Then the set ^ defined in
Section 13.13, Problem 2(b) is also the closure of ^ in the set of the U/ for which /
is ^0, compactly supported and belongs to && . Generalize the result of part (e)
of this problem to the case where /  && is compactly supported and j>0 almost
everywhere. Likewise, generalize part (f ) of the same problem.

(c)    Suppose that X is compact. Then Uf is defined for all functions / e J^jJ(X, ft),
and we have N2(UO ^6xN2(/). If G -/is the class of U7, then G is a continuous
positive self-adjoint operator on LR(X, /it). If F is the closure in LR of Gl/2(LJi) (which
is the orthogonal supplement of Ker(<71/2) = Ker(C)), then the restriction of G112
to F is an isometry of the subspace F of LR onto the Hilbert space H (equipped with
the norm |w|). Hence H = G1/2(L).

(d)    Suppose that X is compact and that the "domination principle" is satisfied,
in the form of (b): that is to say, if / e -5?R is ^0 almost everywhere, and if u e &
is such that Uf(x) :g u(x) almost everywhere in the set of points x where f(x) > 0,
then Uf(x) g u(x) almost everywhere in X. For each A > 0, put R* = G(I + AG)""1.
If /e JR is ^0 almost everywhere, and if g is a function whose class is equal toero on E0, is equal to Srl on EI, and is equal to Sn-j.Snl on  for all